On an elliptic boundary value problem not in divergence form

Abstract:

Let $G$ be a bounded domain in ${R^n}(n \geqslant 2)$ with smooth boundary $\partial G$. We consider the boundary value problem $Mu - cu = f$ on $G$, $u = 0$ on $\partial G$, where $M$ is an elliptic differential operator not in divergence form. We discuss the characterization of the first eigenvalue ${\lambda _0}$ of $M$ and the solvability of the boundary value problem in terms of the relationship between $c( \cdot )$ and ${\lambda _0}$.